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We establish some bounds for the natural distance function on the boundary of such a conformal deformation of the unit ball B, n ≥ 2, that satisfies a Harnack inequality and the condition of the Gehring–Hayman theorem. The construction is useful especially for those points for which the radial limit exists.

A distance estimate on the boundary of a conformal deformation of the unit ball satisfying a Harnack inequality and the Gehring-Hayman property